Exact $1/4$ mean-rung population and $\sqrt{2}$ continuum revival-time ratio in the anti-Jaynes--Cummings vertex at $f_{\star }=\sqrt{\bar{n}+1}$

Abstract

At atomic resonance the Jaynes--Cummings (JC) vertex of a single bosonic mode coupled to a two-level emitter is blind to $f=\omega /\lambda$ at zero detuning and Kerr; the anti-Jaynes--Cummings (AJC) vertex is not. We show that the mean-rung excited population of the AJC block equals exactly $1/4$ at $f_{\star }=\sqrt{\bar{n}+1}$, while the JC population at the same point is $1/2$. The Poisson-averaged packet value at $\bar{n}=16$ is $0.24650$, within $3.5\times 10^{-3}$ of the exact value, with the deviation reproduced by $-\bar{n}/[16(\bar{n}+1{)}^2]$. At $f_{\star }$ the modulation depth of the AJC inversion is $1/2$ and the relative sensitivity of the continuum revival-time ratio $r=t_R^{\mathrm{AJC}}/t_R^{\mathrm{JC}}$ is maximal with value $1/(2\sqrt{\bar{n}+1})$. Full diagonalization on $2(n_{\max }+1)$ states confirms the analytic values. The $1/4$ value can be generated by Floquet-engineered counter-rotating coupling provided that the static transverse rate satisfies $g_0\ll g_1$. Parameter translations for an ideal trapped-ion blue sideband and a flux-tunable circuit-QED coupler are reported.

18 pages, 8 figures